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Sven Herrmann

Fachbereich Mathematik Technische Universität Darmstadt

60th Séminaire Lotharingien de Combinatoire April 2, 2008

Sven Herrmann (TU Darmstadt) Genocchi Numbers SLC 60, April 2, 2008 1 / 16

(2)

Outline

1 f -Vectors of Simplicial Balls

2 Genocchi Numbers

3 Genocchi Numbers and f -Vectors of Simplicial Balls

Sven Herrmann (TU Darmstadt) Genocchi Numbers SLC 60, April 2, 2008 2 / 16

(3)

1 f -Vectors of Simplicial Balls

2 Genocchi Numbers

3 Genocchi Numbers and f -Vectors of Simplicial Balls

Sven Herrmann (TU Darmstadt) Genocchi Numbers SLC 60, April 2, 2008 3 / 16

(4)

Simplicial Balls

B an(n−1)-dimensional simplicial ball i.e. simplicial complex homeomorphic to a ball fi(B)number of i-dimensional faces

boundary∂B of B is a simplicial sphere with face numbers fi(∂B) interior int B of B (not a simplicial complex)

fi(int B) :=fi(B)−fi(∂B)

relation beetween f(∂B)and f(int B)

Sven Herrmann (TU Darmstadt) Genocchi Numbers SLC 60, April 2, 2008 4 / 16

(5)

B an(n−1)-dimensional simplicial ball i.e. simplicial complex homeomorphic to a ball fi(B)number of i-dimensional faces

boundary∂B of B is a simplicial sphere with face numbers fi(∂B) interior int B of B (not a simplicial complex)

fi(int B) :=fi(B)−fi(∂B)

relation beetween f(∂B)and f(int B)

Sven Herrmann (TU Darmstadt) Genocchi Numbers SLC 60, April 2, 2008 4 / 16

(6)

Simplicial Balls

B an(n−1)-dimensional simplicial ball i.e. simplicial complex homeomorphic to a ball fi(B)number of i-dimensional faces

boundary∂B of B is a simplicial sphere with face numbers fi(∂B) interior int B of B (not a simplicial complex)

fi(int B) :=fi(B)−fi(∂B)

relation beetween f(∂B)and f(int B)

Sven Herrmann (TU Darmstadt) Genocchi Numbers SLC 60, April 2, 2008 4 / 16

(7)

B an(n−1)-dimensional simplicial ball i.e. simplicial complex homeomorphic to a ball fi(B)number of i-dimensional faces

boundary∂B of B is a simplicial sphere with face numbers fi(∂B) interior int B of B (not a simplicial complex)

fi(int B) :=fi(B)−fi(∂B)

relation beetween f(∂B)and f(int B)

Sven Herrmann (TU Darmstadt) Genocchi Numbers SLC 60, April 2, 2008 4 / 16

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Simplicial Balls

B an(n−1)-dimensional simplicial ball i.e. simplicial complex homeomorphic to a ball fi(B)number of i-dimensional faces

boundary∂B of B is a simplicial sphere with face numbers fi(∂B) interior int B of B (not a simplicial complex)

fi(int B) :=fi(B)−fi(∂B)

relation beetween f(∂B)and f(int B)

Sven Herrmann (TU Darmstadt) Genocchi Numbers SLC 60, April 2, 2008 4 / 16

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B an(n−1)-dimensional simplicial ball i.e. simplicial complex homeomorphic to a ball fi(B)number of i-dimensional faces

boundary∂B of B is a simplicial sphere with face numbers fi(∂B) interior int B of B (not a simplicial complex)

fi(int B) :=fi(B)−fi(∂B)

relation beetween f(∂B)and f(int B)

Sven Herrmann (TU Darmstadt) Genocchi Numbers SLC 60, April 2, 2008 4 / 16

(10)

Simplicial Balls

B an(n−1)-dimensional simplicial ball i.e. simplicial complex homeomorphic to a ball fi(B)number of i-dimensional faces

boundary∂B of B is a simplicial sphere with face numbers fi(∂B) interior int B of B (not a simplicial complex)

fi(int B) :=fi(B)−fi(∂B)

relation beetween f(∂B)and f(int B)

Sven Herrmann (TU Darmstadt) Genocchi Numbers SLC 60, April 2, 2008 4 / 16

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polytope P: convex hull of finitely many points triangulation T : collection of simplices whose union is P

relation between triangulation of the boundary and T

Sven Herrmann (TU Darmstadt) Genocchi Numbers SLC 60, April 2, 2008 5 / 16

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Example: Triangulated Polytopes

polytope P: convex hull of finitely many points triangulation T : collection of simplices whose union is P

relation between triangulation of the boundary and T

Sven Herrmann (TU Darmstadt) Genocchi Numbers SLC 60, April 2, 2008 5 / 16

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polytope P: convex hull of finitely many points triangulation T : collection of simplices whose union is P

relation between triangulation of the boundary and T

Sven Herrmann (TU Darmstadt) Genocchi Numbers SLC 60, April 2, 2008 5 / 16

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Example: Triangulated Polytopes

simplicial polytope (McMullen 2004)

all faces of P are simplices

boundary is a simplicial (triangulated) sphere

f(∂T)is fixed (if we do not allow additional points) triangulations without interior faces of low dimension combinatorics of unbounded polyhedra (H. & Joswig 2007)

complex of bounded faces is dual to interior of a simplicial ball

special case: tight span of finite metric space (Dress 1984, et al.)

Sven Herrmann (TU Darmstadt) Genocchi Numbers SLC 60, April 2, 2008 6 / 16

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simplicial polytope (McMullen 2004)

all faces of P are simplices

boundary is a simplicial (triangulated) sphere

f(∂T)is fixed (if we do not allow additional points) triangulations without interior faces of low dimension combinatorics of unbounded polyhedra (H. & Joswig 2007)

complex of bounded faces is dual to interior of a simplicial ball

special case: tight span of finite metric space (Dress 1984, et al.)

Sven Herrmann (TU Darmstadt) Genocchi Numbers SLC 60, April 2, 2008 6 / 16

(16)

Example: Triangulated Polytopes

simplicial polytope (McMullen 2004)

all faces of P are simplices

boundary is a simplicial (triangulated) sphere

f(∂T)is fixed (if we do not allow additional points) triangulations without interior faces of low dimension combinatorics of unbounded polyhedra (H. & Joswig 2007)

complex of bounded faces is dual to interior of a simplicial ball

special case: tight span of finite metric space (Dress 1984, et al.)

Sven Herrmann (TU Darmstadt) Genocchi Numbers SLC 60, April 2, 2008 6 / 16

(17)

simplicial polytope (McMullen 2004)

all faces of P are simplices

boundary is a simplicial (triangulated) sphere

f(∂T)is fixed (if we do not allow additional points) triangulations without interior faces of low dimension combinatorics of unbounded polyhedra (H. & Joswig 2007)

complex of bounded faces is dual to interior of a simplicial ball

special case: tight span of finite metric space (Dress 1984, et al.)

Sven Herrmann (TU Darmstadt) Genocchi Numbers SLC 60, April 2, 2008 6 / 16

(18)

Example: Triangulated Polytopes

simplicial polytope (McMullen 2004)

all faces of P are simplices

boundary is a simplicial (triangulated) sphere

f(∂T)is fixed (if we do not allow additional points) triangulations without interior faces of low dimension combinatorics of unbounded polyhedra (H. & Joswig 2007)

complex of bounded faces is dual to interior of a simplicial ball

special case: tight span of finite metric space (Dress 1984, et al.)

Sven Herrmann (TU Darmstadt) Genocchi Numbers SLC 60, April 2, 2008 6 / 16

(19)

simplicial polytope (McMullen 2004)

all faces of P are simplices

boundary is a simplicial (triangulated) sphere

f(∂T)is fixed (if we do not allow additional points) triangulations without interior faces of low dimension combinatorics of unbounded polyhedra (H. & Joswig 2007)

complex of bounded faces is dual to interior of a simplicial ball

special case: tight span of finite metric space (Dress 1984, et al.)

Sven Herrmann (TU Darmstadt) Genocchi Numbers SLC 60, April 2, 2008 6 / 16

(20)

Example: Triangulated Polytopes

simplicial polytope (McMullen 2004)

all faces of P are simplices

boundary is a simplicial (triangulated) sphere

f(∂T)is fixed (if we do not allow additional points) triangulations without interior faces of low dimension combinatorics of unbounded polyhedra (H. & Joswig 2007)

complex of bounded faces is dual to interior of a simplicial ball

special case: tight span of finite metric space (Dress 1984, et al.)

Sven Herrmann (TU Darmstadt) Genocchi Numbers SLC 60, April 2, 2008 6 / 16

(21)

simplicial polytope (McMullen 2004)

all faces of P are simplices

boundary is a simplicial (triangulated) sphere

f(∂T)is fixed (if we do not allow additional points) triangulations without interior faces of low dimension combinatorics of unbounded polyhedra (H. & Joswig 2007)

complex of bounded faces is dual to interior of a simplicial ball

special case: tight span of finite metric space (Dress 1984, et al.)

Sven Herrmann (TU Darmstadt) Genocchi Numbers SLC 60, April 2, 2008 6 / 16

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Dehn-Sommerville Relations

hk =Pk

i=0(−1)k−i nn++1−k1−i

fi, g0=1 and gk =hkhk−1for k ≥1

Theorem (Dehn-Sommerville) S simplicial(n−1)-sphere

hk(S) = hn−k(S) .

Theorem (McMullen & Walkup 1971) B simplicial(n−1)-ball

gk(∂B) = hk(B)−hn−k(B) .

from there one gets relations beetween h(∂B)and h(int B) (McMullen 2005 et al.)

Sven Herrmann (TU Darmstadt) Genocchi Numbers SLC 60, April 2, 2008 7 / 16

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Theorem (Dehn-Sommerville) S simplicial(n−1)-sphere

hk(S) = hn−k(S) .

Theorem (McMullen & Walkup 1971) B simplicial(n−1)-ball

gk(∂B) = hk(B)−hn−k(B) .

from there one gets relations beetween h(∂B)and h(int B) (McMullen 2005 et al.)

Sven Herrmann (TU Darmstadt) Genocchi Numbers SLC 60, April 2, 2008 7 / 16

(24)

Dehn-Sommerville Relations

hk =Pk

i=0(−1)k−i nn++1−k1−i

fi, g0=1 and gk =hkhk−1for k ≥1

Theorem (Dehn-Sommerville) S simplicial(n−1)-sphere

hk(S) = hn−k(S) .

Theorem (McMullen & Walkup 1971) B simplicial(n−1)-ball

gk(∂B) = hk(B)−hn−k(B) .

from there one gets relations beetween h(∂B)and h(int B) (McMullen 2005 et al.)

Sven Herrmann (TU Darmstadt) Genocchi Numbers SLC 60, April 2, 2008 7 / 16

(25)

Theorem (Dehn-Sommerville) S simplicial(n−1)-sphere

hk(S) = hn−k(S) .

Theorem (McMullen & Walkup 1971) B simplicial(n−1)-ball

gk(∂B) = hk(B)−hn−k(B) .

from there one gets relations beetween h(∂B)and h(int B) (McMullen 2005 et al.)

Sven Herrmann (TU Darmstadt) Genocchi Numbers SLC 60, April 2, 2008 7 / 16

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Relations for the f -Vector

Proposition (Klain) S simplicial(n−1)-sphere

fk(S) =

n−1

X

i=k

(−1)i+n−1 i+1

k +1

fi(S)

Proposition (Klain) B simplicial(n−1)-ball

fk(int B) = −fk(∂B)

2 −

n−k−1

X

i=1

(−1)n+k+i 2

k+1+i k+1

fk+i(int B)

Sven Herrmann (TU Darmstadt) Genocchi Numbers SLC 60, April 2, 2008 8 / 16

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S simplicial(n−1)-sphere

fk(S) =

n−1

X

i=k

(−1)i+n−1 i+1

k +1

fi(S)

Proposition (Klain) B simplicial(n−1)-ball

fk(int B) = −fk(∂B)

2 −

n−k−1

X

i=1

(−1)n+k+i 2

k+1+i k+1

fk+i(int B)

Sven Herrmann (TU Darmstadt) Genocchi Numbers SLC 60, April 2, 2008 8 / 16

(28)

Outline

1 f -Vectors of Simplicial Balls

2 Genocchi Numbers

3 Genocchi Numbers and f -Vectors of Simplicial Balls

Sven Herrmann (TU Darmstadt) Genocchi Numbers SLC 60, April 2, 2008 9 / 16

(29)

function

2t et +1 =:

X

n=0

Gntn n! =t+

X

n=1

G2n t2n (2n)! .

The first few numbers are

-1, 1, -3, 17, -155, 2073, -38227, 929569, -28820619, 1109652905

One has

Gn =2(1−22n)Bn, where Bnare theBernoulli numbers.

Sven Herrmann (TU Darmstadt) Genocchi Numbers SLC 60, April 2, 2008 10 / 16

(30)

Genocchi Numbers

Definition

TheGenocchi numbersare defined by means of the generating function

2t et +1 =:

X

n=0

Gntn n! =t+

X

n=1

G2n t2n (2n)! .

The first few numbers are

-1, 1, -3, 17, -155, 2073, -38227, 929569, -28820619, 1109652905

One has

Gn =2(1−22n)Bn, where Bnare theBernoulli numbers.

Sven Herrmann (TU Darmstadt) Genocchi Numbers SLC 60, April 2, 2008 10 / 16

(31)

function

2t et +1 =:

X

n=0

Gntn n! =t+

X

n=1

G2n t2n (2n)! .

The first few numbers are

-1, 1, -3, 17, -155, 2073, -38227, 929569, -28820619, 1109652905

One has

Gn =2(1−22n)Bn, where Bnare theBernoulli numbers.

Sven Herrmann (TU Darmstadt) Genocchi Numbers SLC 60, April 2, 2008 10 / 16

(32)

Genocchi Numbers

Definition

TheGenocchi numbersare defined by means of the generating function

2t et +1 =:

X

n=0

Gntn n! =t+

X

n=1

G2n t2n (2n)! .

The first few numbers are

-1, 1, -3, 17, -155, 2073, -38227, 929569, -28820619, 1109652905

One has

Gn =2(1−22n)Bn, where Bnare theBernoulli numbers.

Sven Herrmann (TU Darmstadt) Genocchi Numbers SLC 60, April 2, 2008 10 / 16

(33)

Genocchi (1817-1889) by Edouard Lucas (1891) investigation goes back to Leonhard Euler

intensively studies by Eric T. Bell (1926, 1929) a lot of generalizations (e.g. Kim et al. 2001, Domaritzki 2004, Zeng &

Zhou 2006)

A001469 in the OEIS

Sven Herrmann (TU Darmstadt) Genocchi Numbers SLC 60, April 2, 2008 11 / 16

(34)

Genocchi Numbers

named after the Italian mathematician Angelo Genocchi (1817-1889) by Edouard Lucas (1891) investigation goes back to Leonhard Euler

intensively studies by Eric T. Bell (1926, 1929) a lot of generalizations (e.g. Kim et al. 2001, Domaritzki 2004, Zeng &

Zhou 2006)

A001469 in the OEIS

Sven Herrmann (TU Darmstadt) Genocchi Numbers SLC 60, April 2, 2008 11 / 16

(35)

Genocchi (1817-1889) by Edouard Lucas (1891) investigation goes back to Leonhard Euler

intensively studies by Eric T. Bell (1926, 1929) a lot of generalizations (e.g. Kim et al. 2001, Domaritzki 2004, Zeng &

Zhou 2006)

A001469 in the OEIS

Sven Herrmann (TU Darmstadt) Genocchi Numbers SLC 60, April 2, 2008 11 / 16

(36)

Genocchi Numbers

named after the Italian mathematician Angelo Genocchi (1817-1889) by Edouard Lucas (1891) investigation goes back to Leonhard Euler

intensively studies by Eric T. Bell (1926, 1929) a lot of generalizations (e.g. Kim et al. 2001, Domaritzki 2004, Zeng &

Zhou 2006)

A001469 in the OEIS

Sven Herrmann (TU Darmstadt) Genocchi Numbers SLC 60, April 2, 2008 11 / 16

(37)

Genocchi (1817-1889) by Edouard Lucas (1891) investigation goes back to Leonhard Euler

intensively studies by Eric T. Bell (1926, 1929) a lot of generalizations (e.g. Kim et al. 2001, Domaritzki 2004, Zeng &

Zhou 2006)

A001469 in the OEIS

Sven Herrmann (TU Darmstadt) Genocchi Numbers SLC 60, April 2, 2008 11 / 16

(38)

Combinatorical Interpretation

combinatorial interpretation due to Dominque Dumont (1974):

absolute value of G2n+2equals the number of permutationsτ of {1,2, . . . ,2n}such thatτ(i)>i if and only if i is odd.

n=2:

1 2 3 4 2 1 4 3

1 2 3 4 3 1 4 2

1 2 3 4 3 2 4 1

=⇒ |G6|=3

Sven Herrmann (TU Darmstadt) Genocchi Numbers SLC 60, April 2, 2008 12 / 16

(39)

absolute value of G2n+2equals the number of permutationsτ of {1,2, . . . ,2n}such thatτ(i)>i if and only if i is odd.

n=2:

1 2 3 4 2 1 4 3

1 2 3 4 3 1 4 2

1 2 3 4 3 2 4 1

=⇒ |G6|=3

Sven Herrmann (TU Darmstadt) Genocchi Numbers SLC 60, April 2, 2008 12 / 16

(40)

Combinatorical Interpretation

combinatorial interpretation due to Dominque Dumont (1974):

absolute value of G2n+2equals the number of permutationsτ of {1,2, . . . ,2n}such thatτ(i)>i if and only if i is odd.

n=2:

1 2 3 4 2 1 4 3

1 2 3 4 3 1 4 2

1 2 3 4 3 2 4 1

=⇒ |G6|=3

Sven Herrmann (TU Darmstadt) Genocchi Numbers SLC 60, April 2, 2008 12 / 16

(41)

absolute value of G2n+2equals the number of permutationsτ of {1,2, . . . ,2n}such thatτ(i)>i if and only if i is odd.

n=2:

1 2 3 4 2 1 4 3

1 2 3 4 3 1 4 2

1 2 3 4 3 2 4 1

=⇒ |G6|=3

Sven Herrmann (TU Darmstadt) Genocchi Numbers SLC 60, April 2, 2008 12 / 16

(42)

Recursion Formulae

There are a lot of formulae to compute Gn

Proposition

G2n =−n−1 2

n−1

X

k=1

2n 2k

G2k and G2n =−1−

n−1

X

k=1

2n 2k−1

G2k 2k Lemma

For n2 we have 2n−1

n G2n =−1 2

n−1

X

k=1

2n−1 2k −1

2k−1 k G2k

Sven Herrmann (TU Darmstadt) Genocchi Numbers SLC 60, April 2, 2008 13 / 16

(43)

Proposition

G2n =−n−1 2

n−1

X

k=1

2n 2k

G2k and G2n =−1−

n−1

X

k=1

2n 2k−1

G2k 2k Lemma

For n2 we have 2n−1

n G2n =−1 2

n−1

X

k=1

2n−1 2k −1

2k−1 k G2k

Sven Herrmann (TU Darmstadt) Genocchi Numbers SLC 60, April 2, 2008 13 / 16

(44)

Recursion Formulae

There are a lot of formulae to compute Gn

Proposition

G2n =−n−1 2

n−1

X

k=1

2n 2k

G2k and G2n =−1−

n−1

X

k=1

2n 2k−1

G2k 2k Lemma

For n2 we have 2n−1

n G2n =−1 2

n−1

X

k=1

2n−1 2k −1

2k−1 k G2k

Sven Herrmann (TU Darmstadt) Genocchi Numbers SLC 60, April 2, 2008 13 / 16

(45)

1 f -Vectors of Simplicial Balls

2 Genocchi Numbers

3 Genocchi Numbers and f -Vectors of Simplicial Balls

Sven Herrmann (TU Darmstadt) Genocchi Numbers SLC 60, April 2, 2008 14 / 16

(46)

Main Theorem

Theorem (H. 2007)

For each simplicial(n−1)-ball B and n−k even we have

fk(int B) =

n2k

X

i=1

G2i 2i

k+2i−1 k+1

fk+2i−2(∂B)

k +2i k+1

fk+2i−1(int B)

! .

fk for nk odd can be computed from the even ones

fk(int B)depending an⌊(n−k)/2⌋fi(int B)

Sven Herrmann (TU Darmstadt) Genocchi Numbers SLC 60, April 2, 2008 15 / 16

(47)

For each simplicial(n−1)-ball B and n−k even we have

fk(int B) =

n2k

X

i=1

G2i 2i

k+2i−1 k+1

fk+2i−2(∂B)

k +2i k+1

fk+2i−1(int B)

! .

fk for nk odd can be computed from the even ones

fk(int B)depending an⌊(n−k)/2⌋fi(int B)

Sven Herrmann (TU Darmstadt) Genocchi Numbers SLC 60, April 2, 2008 15 / 16

(48)

Main Theorem

Theorem (H. 2007)

For each simplicial(n−1)-ball B and n−k even we have

fk(int B) =

n2k

X

i=1

G2i 2i

k+2i−1 k+1

fk+2i−2(∂B)

k +2i k+1

fk+2i−1(int B)

! .

fk for nk odd can be computed from the even ones

fk(int B)depending an⌊(n−k)/2⌋fi(int B)

Sven Herrmann (TU Darmstadt) Genocchi Numbers SLC 60, April 2, 2008 15 / 16

(49)

nk even and ke

n2k

X

i=1

G2i 2i

k+2i−1 k+1

fk+2i−2(∂B) =

n2k

X

i=1

G2i 2i

k+2i k +1

fk+2i−1(int B)

⌊(e+2)/2⌋

⌊(e+3)/2⌋

equations

⌊(n−1−e)/2⌋(

⌊(n−e)/2⌋

unknowns fk(int B)

similar equations for the h-vector computed and used: H. &

Joswig 2007

Sven Herrmann (TU Darmstadt) Genocchi Numbers SLC 60, April 2, 2008 16 / 16

(50)

Simplicial Balls without Small Interior Faces

B a simplical(n−1)-ball without interior faces of dimension≤e nk even and ke

n2k

X

i=1

G2i 2i

k+2i−1 k+1

fk+2i−2(∂B) =

n2k

X

i=1

G2i 2i

k+2i k +1

fk+2i−1(int B)

⌊(e+2)/2⌋

⌊(e+3)/2⌋

equations

⌊(n−1−e)/2⌋(

⌊(n−e)/2⌋

unknowns fk(int B)

similar equations for the h-vector computed and used: H. &

Joswig 2007

Sven Herrmann (TU Darmstadt) Genocchi Numbers SLC 60, April 2, 2008 16 / 16

(51)

nk even and ke

n2k

X

i=1

G2i 2i

k+2i−1 k+1

fk+2i−2(∂B) =

n2k

X

i=1

G2i 2i

k+2i k +1

fk+2i−1(int B)

⌊(e+2)/2⌋

⌊(e+3)/2⌋

equations

⌊(n−1−e)/2⌋(

⌊(n−e)/2⌋

unknowns fk(int B)

similar equations for the h-vector computed and used: H. &

Joswig 2007

Sven Herrmann (TU Darmstadt) Genocchi Numbers SLC 60, April 2, 2008 16 / 16

(52)

Simplicial Balls without Small Interior Faces

B a simplical(n−1)-ball without interior faces of dimension≤e nk even and ke

n2k

X

i=1

G2i 2i

k+2i−1 k+1

fk+2i−2(∂B) =

n2k

X

i=1

G2i 2i

k+2i k +1

fk+2i−1(int B)

⌊(e+2)/2⌋

⌊(e+3)/2⌋

equations

⌊(n−1−e)/2⌋(

⌊(n−e)/2⌋

unknowns fk(int B)

similar equations for the h-vector computed and used: H. &

Joswig 2007

Sven Herrmann (TU Darmstadt) Genocchi Numbers SLC 60, April 2, 2008 16 / 16

(53)

nk even and ke

n2k

X

i=1

G2i 2i

k+2i−1 k+1

fk+2i−2(∂B) =

n2k

X

i=1

G2i 2i

k+2i k +1

fk+2i−1(int B)

⌊(e+2)/2⌋

⌊(e+3)/2⌋

equations

⌊(n−1−e)/2⌋(

⌊(n−e)/2⌋

unknowns fk(int B)

similar equations for the h-vector computed and used: H. &

Joswig 2007

Sven Herrmann (TU Darmstadt) Genocchi Numbers SLC 60, April 2, 2008 16 / 16

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A class F of real or complex valued functions is said to be inverse closed if 1/f remains in the class whenever f is in the class and it does not vanish, and it is said to